TITLE: Finite Powers of ordered topological spaces
QUESTION [6 upvotes]: Given a totally ordered set $I$.
It is well known that $I$ becomes a normal Hausdorff space when endowed with the order topology.
What can be said about finite products
$I \times \cdots \times I$ ? When are they normal?
Are there sufficient and necessary conditions on the order of $I$ for all finite powers to be normal?

REPLY [4 votes]: To answer Will Brian's question, notice that the product space $(\omega_1 + 1) \times \omega_1$ is not normal.  (This fact is well-known; the Pressing Down Lemma implies that the diagonal and the right edge cannot be completely separated.) Let $I$ be the ordinal space $\omega_1 + \omega_1$.  Since $I$ contains each of the spaces $\omega_1 + 1$ and $\omega_1$ as a closed subspace, the product $I \times I$ is not normal.